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Crossed products as compact quantum metric spaces

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators Infinite-dimensional manifolds

Published online by Cambridge University Press:  19 December 2024

Mario Klisse Open the ORCID record for Mario Klisse [Opens in a new window]
Mario Klisse*
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium
*
e-mail: mario.klisse@kuleuven.be
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Abstract

By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C$^\ast $-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.

MSC classification

Primary: 46L05: General theory of $C^*$-algebras 46L87: Noncommutative differential geometry
Secondary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 47L65: Crossed product algebras (analytic crossed products) 58B34: Noncommutative geometry (à la Connes)
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 31
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The author is supported by FWO research project G090420N of the Research Foundation Flanders.

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